It’s Bugging Me To No End
So throbbin asks me to check out a question he found on a disussion forum somwhere, and it’s been bugging me to no end…..
You’ve been wandering in the desert for days, you’re dying of thirst, you need water, you come across a village where you (somehow) know that half the people will lie to you and the other half will tell you the truth, you can’t tell them apart, and you only have the strength to ask one question about where to get water before you croak – you’re life depends on it, what question will you ask?
The Old Mormon…
Wasn’t there a very similar question in Labrynth? Two talking doors, one of which is the right way, one alway lies, one always tells the truth…damn it…now I’m going to have to go and rent Labrynth…
The question is:
“If I were to ask you where I can find water, what would you say?”
Well, that uses up your one question, but how does it get you any closer to the water?
If the person answers with, “I’d say it was over there” you’re still no further ahead than when you asked – maybe they told the truth, maybe they lied.
I don’t think there is a single question.
If it was a binary choice ( eg: does the well on the right or the well on the left have water ) you could ask: if I were you which well would I say has the water in it.
I don’t think it works with an open ended question.
The basic question is: Where is the water?
There are two possible responses. The truthteller will tell the truth, the liar will lie.
If asked, “If I were to ask you where to find water, what would you say”:
a) The truthteller will tell the truth about what she would say: that equals the truth.
b) The liar will lie about lying: that equals the trutn.
You could also ask one person what the other would say. Some people find that logic easier to follow. The truthteller will say that the liar would point to the wrong door, the liar would say the truthteller would point to the wrong door. Take the other door.
Or you could just say…”hey, let me buy you a drink.” Even a liar would perk up and lead you to the nearest bar at the offer of a free pint.
balbulican, the problem is you don’t know whether you are asking the truth teller or the liar.
PS: I think your other plan has a lot of merit.
STageleft, it doesn’t matter which one you ask, that’t the point: by framing the question in a question, you compel the liar to tell you the truth = eg., to lie about his lie.
Dammit, I must start reading the prior comments before I make my own. [blushes]
Cottonwood has it.
“If you were one of the others, which way would you point to show me to the water?”
Then take the opposite path.
The Cottonwood/balbulican solutions are exactly the same: in both cases you frame the question so that, in effect, the truth teller reponds with a double positive, while the liar is forced to respond with a double negative (i.e., a positive).
As B notes, the key is to frame the question within a question and include one of the possible answers. You have to be dealing with a binary choice (and in the problem as presented will generally be faced with people who answer only “Yes” or “No”).
First frame the decision; for example, “X is true”, as in “I should take the left-hand fork in the road”, “I should open the right-hand door”, “I should go to that well if I wish to obtain water”.
If you ask, “Is X true?” and X is in fact true, the truth-teller will answer “Yes” and the liar will answer “No”, and vice versa if X is in fact false.
But, if you ask, “If I ask you whether X is true, what will you answer?” (or “…will you answer Yes?), then:
1) The truth-teller must say “Yes” if his answer to “Is X true?” would be “Yes” and “No” otherwise – his answer is the same to the direct or indirect question.
2) The liar must say “Yes” if his answer to “Is X true?” would be “No” (ie. if X is true) and “No” otherwise.
In both cases, “Yes” is the answer elicited if X is true and “No” if X is false, so you know whether X is true.
That was all very interesting. I appreciate IrC’s explanation with the use of X. I understand X.
I feel I can contribute by noting, if i remember correctly, that the character in Labyrinth actually chose incorrectly. The door she chose, after using what she thought was flawless reasoning, led her to a fall down a long tunnel of grabbing hands.